Magic: the Gathering is as Hard as Arithmetic
This provides foundational theoretical insights into the complexity of a popular game, advancing understanding of computational limits in game theory, though it is incremental by building on prior work.
The paper tackles the computational complexity of optimal play in Magic: the Gathering, showing that the 'mate-in-n' problem is Δ^0_n-hard and that optimal play is non-arithmetic, using standard tournament-legal decks without relying on stochasticity or hidden information.
Magic: the Gathering is a popular and famously complicated card game about magical combat. Recently, several authors including Chatterjee and Ibsen-Jensen (2016) and Churchill, Biderman, and Herrick (2019) have investigated the computational complexity of playing Magic optimally. In this paper we show that the ``mate-in-$n$'' problem for Magic is $Δ^0_n$-hard and that optimal play in two-player Magic is non-arithmetic in general. These results apply to how real Magic is played, can be achieved using standard-size tournament legal decks, and do not rely on stochasticity or hidden information. Our paper builds upon the construction that Churchill, Biderman, and Herrick (2019) used to show that this problem was at least as hard as the halting problem.